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Quiz: Introduction to Functions

Test your understanding of functions, function notation, domain and range, and types of functions with these questions.

1. Which statement best defines a function?

  1. A relationship where each output has exactly one input
  2. A relationship where each input has exactly one output
  3. Any set of ordered pairs
  4. A relationship where some inputs can have multiple outputs
Show Answer

The correct answer is B. A function is a special type of relation where each input value (x) corresponds to exactly one output value (y). This is the fundamental definition of a function. Option A reverses the relationship. Option C is too broad—it describes any relation, not specifically a function. Option D contradicts the definition of a function.

Concept Tested: Function

See: What is a Function?

2. For the function f(x) = 3x - 5, what is f(4)?

  1. 12
  2. -2
  3. 17
  4. 7
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The correct answer is D. Substitute 4 for x: f(4) = 3(4) - 5 = 12 - 5 = 7. Function notation f(4) asks "what is the output when the input is 4?" Option A shows only 3×4 without subtracting 5. Option B uses -4 instead of 4. Option C adds instead of subtracts.

Concept Tested: Function Notation / Evaluating Functions

See: Evaluating Functions

3. Which set of ordered pairs represents a function?

  1. {(1, 2), (2, 3), (1, 4), (3, 5)}
  2. {(1, 2), (2, 2), (3, 2), (4, 2)}
  3. {(1, 2), (2, 3), (3, 4), (2, 5)}
  4. {(5, 1), (5, 2), (5, 3), (5, 4)}
Show Answer

The correct answer is B. In this set, each input (1, 2, 3, 4) has exactly one output (all map to 2). This satisfies the definition of a function. Option A has input 1 mapping to both 2 and 4, violating the function rule. Option C has input 2 mapping to both 3 and 5. Option D has input 5 mapping to four different outputs. Multiple inputs can share the same output, but each input must have only one output.

Concept Tested: Relation / Function Definition

See: Relations and Functions

4. What is the domain of the function f(x) = √(x - 3)?

  1. All real numbers
  2. x > 3
  3. x < 3
  4. x ≥ 3
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The correct answer is D. The domain consists of all input values for which the function is defined. Since you cannot take the square root of a negative number (in the real number system), we need x - 3 ≥ 0, which means x ≥ 3. Option A ignores the restriction. Option B uses strict inequality when x = 3 is actually allowed. Option C reverses the inequality direction.

Concept Tested: Domain

See: Domain and Range

5. Which graph fails the vertical line test and therefore is NOT a function?

  1. A parabola opening upward
  2. A circle
  3. A straight line with positive slope
  4. An exponential curve
Show Answer

The correct answer is B. A circle fails the vertical line test because a vertical line can intersect a circle at two points, meaning one input (x-value) would correspond to two outputs (y-values). This violates the definition of a function. Options A, C, and D all pass the vertical line test—any vertical line intersects each of these graphs at most once.

Concept Tested: Vertical Line Test

See: Vertical Line Test

6. If f(x) = x² and g(x) = x + 3, what is (f + g)(2)?

  1. 9
  2. 25
  3. 5
  4. 11
Show Answer

The correct answer is A. Function addition works by adding the outputs: (f + g)(2) = f(2) + g(2). First, f(2) = 2² = 4. Then g(2) = 2 + 3 = 5. So (f + g)(2) = 4 + 5 = 9. Option B calculates (f ∘ g)(2) = f(g(2)) = f(5) = 25. Option C gives only g(2). Option D incorrectly adds the functions algebraically first.

Concept Tested: Function Operations

See: Function Operations

7. Which type of function shows a constant rate of change?

  1. Quadratic function
  2. Exponential function
  3. Linear function
  4. Absolute value function
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The correct answer is C. Linear functions have a constant rate of change (the slope m in y = mx + b). For every unit increase in x, y changes by the same amount. Quadratic functions have increasing/decreasing rates of change. Exponential functions have rates that change multiplicatively. Absolute value functions have different rates on each side of the vertex.

Concept Tested: Linear Function / Rate of Change

See: Types of Functions

8. What is the range of the function f(x) = x² for all real numbers x?

  1. All real numbers
  2. x ≥ 0
  3. y ≥ 0
  4. y ≤ 0
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The correct answer is C. The range is the set of all possible output values. Since x² is always non-negative for any real number x, the output values are all y ≥ 0. The smallest value is 0 (when x = 0), and values can be arbitrarily large. Option A ignores that squares are non-negative. Option B confuses domain with range. Option D reverses the inequality.

Concept Tested: Range

See: Domain and Range

9. Which represents the independent variable in a function?

  1. The output value
  2. The y-coordinate
  3. The input value
  4. The range
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The correct answer is C. The independent variable is the input to the function, typically represented by x. It's called "independent" because you can choose its value freely. The dependent variable (output/y-coordinate) depends on what input you choose. Option A describes the dependent variable. Option B is another name for the output. Option D is the set of all outputs.

Concept Tested: Independent Variable / Dependent Variable

See: Function Notation

10. A piecewise function is defined as f(x) = {x + 1 if x < 0; 2x if x ≥ 0}. What is f(-3) + f(3)?

  1. 3
  2. 4
  3. 8
  4. -2
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The correct answer is B. For f(-3), since -3 < 0, we use the first rule: f(-3) = -3 + 1 = -2. For f(3), since 3 ≥ 0, we use the second rule: f(3) = 2(3) = 6. Therefore, f(-3) + f(3) = -2 + 6 = 4. Option A uses only f(3). Option C uses the wrong rule for f(-3). Option D gives only f(-3).

Concept Tested: Piecewise Function

See: Piecewise Functions